TIFR CSE 2018 | Part A | Question: 14

Question

Let $A$ be an $n\times n$ invertible matrix with real entries whose row sums are all equal to $c$. Consider the following statements:

1. Every row in the matrix $2A$ sums to $2c$.
2. Every row in the matrix $A^{2}$ sums to $c^{2}$.
3. Every row in the matrix $A^{-1}$ sums to $c^{-1}$.

Which of the following is TRUE?

• A. none of the statements $(1), (2), (3)$ is correct
• B. statement $(1)$ is correct but not necessarily statements $(2)$ or $(3)$
• C. statement $(2)$ is correct but not necessarily statements $(1)$ or $(3)$
• D. statement $(1)$  and $(2)$ are correct but not necessarily statement $(3)$
• E. all the three statements $(1), (2),$ and $(3)$ are correct

Comments

yes..

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All the prosperity are satisfied but by taking example, any mathematical way to do it ?

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i solved it using abcd as values and applying condition ad-bc !=0 and a+b=c+d='c'

Answer 1

Take a matrix:

$\begin{bmatrix} a & b\\ b& a \end{bmatrix}$

Sum of each row$= a+b$ [ matrix is invertible hence determinant is not $0$ i.e. $a$ and $b$ are different]

Now lets prove: 

$2\times \begin{bmatrix} a & b\\ b& a \end{bmatrix} = \begin{bmatrix} 2a & 2b\\ 2b& 2a \end{bmatrix}:$  Sum of each row $= 2(a+b)$

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$\begin{bmatrix} a & b\\ b& a \end{bmatrix}\times \begin{bmatrix} a & b\\ b& a \end{bmatrix} = \begin{bmatrix} a^{2} +b^{2} & ab +ba\\ ab +ba & a^{2} +b^{2} \end{bmatrix}:$  Sum of each rows = $(a^2+b^2+2ab)= (a+b)^2$

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$\frac{1}{a^{2}-b^{2}}\begin{bmatrix} a & -b\\ -b& a \end{bmatrix}= \frac{(a-b)}{a^{2}-b^{2}} = \frac{(a-b)}{(a+b) (a-b)}= \frac{1}{a+b}$

Hence, all are true

Correct Answer: $E$

Comments

@Doraemon

In your example matrix the first and second row do not sum to "c".

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For this question i have taken a matrix [[1,1],[0,2]].

Only statement 1 and 2 are verifying but 3 is not verifying. Pleade check once

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I checked statement 3 is also verifying for your given matrix

Answer 2

Proof: https://math.stackexchange.com/a/3428371/545704

Comments

Gold

Answer 3

If one of the eigen value of matrix A is c then one of eigen value of matrix 2A is 2c.

Similarly for matrix A^2 and A^(-1).

Hence Option(E) should be the answer.

Answer 4

Simplest way is to just take an identity matrix and to check for all options , if some option is true it will be also true for identity matrix ;p so answere will easily come out to be E)